Theorem opnnei 23030

Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)

Assertion

Ref	Expression
opnnei	⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei

Step	Hyp	Ref	Expression
1		0opn 22814	. . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
2	1	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3		eleq1 2819	. . . . 5 ⊢ (𝑆 = ∅ → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
4	3	adantl 481	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
5	2, 4	mpbird 257	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆 ∈ 𝐽)
6		rzal 4454	. . . 4 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
7	6	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
8	5, 7	2thd 265	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9		opnneip 23029	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
10	9	3expia 1121	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑥 ∈ 𝑆 → 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
11	10	ralrimiv 3123	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
12	11	ex 412	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
13	12	adantr 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14		df-ne 2929	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15		r19.2z 4440	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
16	15	ex 412	. . . . . 6 ⊢ (𝑆 ≠ ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
17	14, 16	sylbir 235	. . . . 5 ⊢ (¬ 𝑆 = ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18		eqid 2731	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	neii1 23016	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ⊆ ∪ 𝐽)
20	19	ex 412	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
21	20	rexlimdvw 3138	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
22	17, 21	sylan9r 508	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
23	18	ntrss2 22967	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
24	23	adantr 480	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
26	25	snss 4732	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
27	26	ralbii 3078	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28		dfss3 3918	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
29	28	biimpri 228	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
30	29	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
31	27, 30	sylan2br 595	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
32	24, 31	eqssd 3947	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
33	32	ex 412	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
34	25	snss 4732	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 ↔ {𝑥} ⊆ 𝑆)
35		sstr2 3936	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊆ 𝑆 → (𝑆 ⊆ ∪ 𝐽 → {𝑥} ⊆ ∪ 𝐽))
36	35	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ∪ 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
38	34, 37	biimtrid 242	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
39	38	imp 406	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → {𝑥} ⊆ ∪ 𝐽)
40	18	neiint 23014	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41	40	3com23 1126	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42	41	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
43	39, 42	syldan 591	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
44	43	ralbidva 3153	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
45	18	isopn3 22976	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
46	33, 44, 45	3imtr4d 294	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
47	46	ex 412	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽)))
48	47	com23 86	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
49	48	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
50	22, 49	mpdd 43	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
51	13, 50	impbid 212	. 2 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
52	8, 51	pm2.61dan 812	1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  ∅c0 4278  {csn 4571  ∪ cuni 4854  ‘cfv 6476  Topctop 22803  intcnt 22927  neicnei 23007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-top 22804  df-ntr 22930  df-nei 23008

This theorem is referenced by:  neiptopreu  23043  flimcf  23892